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It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself pl

It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matrices and Gaussian elimination, all the while bored out of your skull, until you have the tools to really start to approach the interesting areas. Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks. This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. It's funny because it's just a pedagogical nightmare to get students to care about any of this until one day two years later it all just makes sense.

Chinjut 1y ago
Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".
- andrewla 1y ago
  Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". I stuck to it because that was what the course taught, and they wouldn't teach me multidimensional calculus without making me learn this stuff first, but it was months and years later when I suddenly found that linear functions were everywhere and I somehow magically had the tools and the knowledge to do stuff with them.
  - JadeNB 1y ago
    > But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". This seems to make it good motivation for an intellectually curious student—"linear functions are the simplest, dumbest kind of function, and yet they still teach us this new and exotic kind of multiplication." That's not how I learned it (I was the kind of obedient student who was interested in a mathematical definition because I was told that I should be), but I can't imagine that I wouldn't have been intrigued by such a presentation!
  - ndriscoll 1y ago
    Linear functions are the ones that we can actually wrap our heads around (maybe), and the big trick we have to understand nonlinear problems is to use calculus to be able to understand them in terms of linear ones again. Problems that can't be made linear tend to be exceptionally difficult, so basically any topic you want to learn is going to be calculus+linear algebra because everything else is too hard. The real payoff though is after you do a deep dive and convince yourself there's plenty of theory and all of these interesting examples and then you learn about SVD or spectral theorems and that when you look at things correctly, you see they act independently in each dimension by... just multiplication by a single number. Unclear whether to be overwhelmed or underwhelmed by the revelation. Or perhaps a superposition.
  - bananaflag 1y ago
    Yeah, concepts can make a student reject them with passion. I remember in a differential geometry course, when we reached "curves on surfaces", I thought "what stupidity! what are the odds a curve lies exactly on a surface?"
- cassepipe 1y ago
  Agree.The fact that it's just linear functions is what made it click for me
msgodel 1y ago
I don't know, maybe it's because I read the book on the side in highschool when I was supposed to be doing something else but I really loved Linear Algebra. Once I understood what it was I used matrix operations for everything. Vector spaces are such a powerful abstraction.
srean 1y ago
> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these follow the definition of a linear transformation. That rotating a sum is same as summing the rotated(s). [One may also mention that all differentiable functions (from vector to vector) are locally linear.] Then you define adding two linear transformations using vector addition. Next you can define scaling a linear transformation. The point being that the combination can be expressed as linear transformations themself. No need to represent the vectors as R^d, geometric arrows and parallelogram rule would suffice. Finally, one demonstrates composition of linear transformations and the fact that the result itself is a linear transformation. The beautiful reveal is that this addition and composition of linear transformations behave almost the same as addition and multiplication of real numbers. The addition asociates and commutes. The multiplication associates but doesn't necessarily commute. Most strikingly, the operations distributes. It's almost like algebra of real numbers ! Now, when you impose a coordinate system or choose a basis, the students can discover that matrix multiplication rule for themselves over a couple of days of playing with it -- Look, rather than maintaining this long list of linear transformations, I can store it as a single linear transformation in the chosen basis.
- cosmic_cheese 1y ago
  If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.
  - te 1y ago
    codingthematrix.com
  - viewtransform 1y ago
dapper_bison17 1y ago
> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. Which books or “non-standard” resources would you recommend then, that do a better job?
- andrewla 1y ago
  I have yet to encounter an approach that is not boring. You just have to power through it. This approach seems as good as any. Once you get to eigenvalues (in my opinion) things start to pick up in terms of seeing that linear spaces are actually interesting. This approach sort of betrays itself when the very first section about scalars has this line: > Vectors are often written vertically in column form, which emphasizes their role in matrix multiplication: This is a big "what?" moment because we don't know why we should care about anything in that sentence. Just call it a convention and later on we can see its utility.
  - _mu 1y ago
    Maybe we can petition Paul Lockhart to do a book on Linear Algebra, I would definitely buy it.
Sharlin 1y ago
For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)
- andrewla 1y ago
  To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.
  - Sharlin 1y ago
    I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.
    - andrewla 1y ago
      Maybe I was making unwarranted assumptions about the nature of your way to learn linear algebra. The approaches that I've seen invariably have to produce a sample matrix, and rotation is really the best example. The rotation matrix is going to have sines and cosines, and understanding what that means is not trivial; and even now if you asked me to write a rotation matrix I would have to work it out from scratch. Easy enough to do mechanically but I have no intuitions here even now.
      - Sharlin 1y ago
        Rotation matrices are somewhat mysterious to the uninitiated, but so is matrix multiplication until it "clicks". Whether it ever clicks is a function of the quality of the learning resource (I certainly do not recommend trying to learn linalg via 3D graphics by just dabbling without a good graphics-oriented textbook or tutorial – that usually doesn’t end well). Anyway, I believe that it's perfectly possible to explain rotation matrices so that it "clicks" with a high probability, as long as you understand the basic fact that (cos a, sin a) is the point that you get when you rotate the point (1, 0) by angle a counter-clockwise about the origin (that's basically their definition!) Involving triangles at all is fully optional.
deleted 1y ago
[deleted]
tptacek 1y ago
I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just have it. I learned from Strang, for what it's worth, which is basically LU, spaces, QR, then spectral. I am really bad at math, for what it's worth; this is just the one advanced math subject that intuitively clicked for me.
- andrewla 1y ago
  I haven't looked at Strang's approach. The "x = b / A" is a bit of a gut-punch on first look because my mind immediately tells me all the ways that that does not work. It makes a some sense once I take a second to think about it, and I can see why it would make you want to jump in a little deeper, but matrices being non-commutative makes me cringe at the idea of a division operator which does not very very clearly spell out where it appears in the chain. Ax = b is all well and good, but AxA^-1 = bA^-1 is not meaningful; the application/composition order is very important.
  - deleted 1y ago
    [deleted]
- dpflan 1y ago
  MIT OCW is an amazing resource -- anyone can learn from Strang, which is a goldmine. He also created a course on using Linear Algebra for machine learning: > Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning. - MIT OCW Course: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/ https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-an...) - The text book website: Linear Algebra and Learning from Data (2019) https://math.mit.edu/~gs/learningfromdata/ https://math.mit.edu/~gs/learningfromdata/ - The Classic Linear Algebra Course: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...
winwang 1y ago
I hated precalc, but I loved proof-based linear algebra, my first university math course (linalg + multivariate calc). The simplicity(/beauty) of matrix multiplication still irks me though, in the sense of "wow, seriously? when you work it out, it really looks that simple?"
jcranmer 1y ago
> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix as a description of how to convert one sense of basis vectors to another set of basis vectors, and for that, you first need to understand how vectors are described in terms of basis vectors.
- nh23423fefe 1y ago
  I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.
  - jcranmer 1y ago
    If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space. If I write a matrix, say, this: [[1 2] [3 4] [5 6]] What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first vector space are represented as a linear combination of the basis vectors of the second vector space. Of course, the transformed vectors may not necessarily be a basis of the latter vector space. > The natural motivation of matrices is as representing systems of equations. That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.
    - nh23423fefe 1y ago
joshmarlow 1y ago
The older I get the more convinced I am that "math is not hard; teaching math is hard".
- rramadass 1y ago
  This is far more truer than most people may realize. Because there is so much to teach/learn, "Modern Mathematics" syllabi has devolved into giving students merely an exposure to all possible mathematical tools in an abstract manner, dis-jointly with no unifying framework, and no motivating examples to explain the need for such mathematics. Most teachers are parrots and have no understanding/insight that they can convey to students and so the system perpetuates itself in a downward spiral. The way to properly teach/learn mathematics is to follow V.I.Arnold's advice i.e. On Teaching Mathematics - https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-... Ground all teaching in actual physical phenomena (in the sense of existence with a purpose) and then show the invention/derivation of abstract mathematics to explain such phenomena. Everything is "Applied Mathematics", there is no "Pure Mathematics" which is just another name for "Abstract Mathematics" to generalize methods of application to different and larger classes of problems.
- gsinclair 1y ago
  As a maths teacher who is interested in (and sufficiently skilled at) programming, I find teaching programming to be very hard, even to interested students. Teaching maths to interested students is not hard (for me).
jameshart 1y ago
What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices. Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.
- JadeNB 1y ago
  I think that, to be frank, it's a combination of (1) a curriculum developed before it was clear how ubiquitous linear algebra would become, and (2) the fact that it's a lot easier to come up with a standardized assessment for algorithmic calculus than for linear algebra, precisely because linear algebra is both conceptual and proof-based in a way that has been squeezed out of algorithmic calculus. (I use algorithmic calculus to describe the high-school subject, and distinguish it from what in American universities is usually called "analysis," where one finally has the chance to make the acquaintance of the conceptual and proof-based aspects squeezed out of algorithmic calculus.)
- bee_rider 1y ago
  I think it was a brilliant and evil trick by the linear algebra folks. Start the path at calculus. Naturally, this will lead to differential equations. Trick the engineers into defining everything in terms of differential equations. The engineers will get really annoyed, because solving differential equations is impossible. Then, the mathematicians swoop in with the idea of discretizing everything and using linear algebra to step through it instead. Suddenly they can justify all the million-by-millions matrices they wanted and everybody thinks they are heroes. Engineers will build the giant vector processing machines that they want.
  - rramadass 1y ago
    Ha, Ha, True dat :-)
- ViscountPenguin 1y ago
  That's very strange, where I live linear algebra was a significant portion of the highschool maths curriculum. The actual presentation was terrible, I'll be lucky if I die before having to invert a matrix by hand again, but it was there.
BenFranklin100 1y ago
Probability and statistics falls into that category too. It’s one of the more boring undergraduate math courses, but is mind-boggling useful in the real world. (Basic probability / combinatorics is actually pretty cool, but both tend to be glossed over.)
Sleaker 1y ago
I went through Khan academy on linear algebra a long time ago because I wanted to learn how to write rendering logic, for me I was implementing things as I learned them with immediate feedback for a lot of the material. Was probably the single most useful thing I learned then.
ViscountPenguin 1y ago
I found the university level presentation of "Vector spaces -> Linear functions -> Matrices are isomorphic to linear functions" much more motivating than the rote mechanics I was taught in highschool, but it's hard to see if I would've had that appreciation without being taught the shitty way first.
127 1y ago
I had the opposite experience when learning linear algebra as I was also doing 3D computer graphics at the time. It was super interesting and fun. I guess you just have to find an application for it.
hiAndrewQuinn 1y ago
Linear algebra was such a tedious thing to learn I skipped over it to abstract algebra and doubled back once I had some kind of minimally interesting framework to work with it against. Normally I think this is a foolish way to do things, but sometimes things are just so dull you have to take the hard route to power through at all.
jonahx 1y ago
> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. Short, simple answer to that question by Michael Penn: https://www.youtube.com/watch?v=cc1ivDlZ71U https://www.youtube.com/watch?v=cc1ivDlZ71U Another interesting treatment by Math the World: https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s There's no impenetrable mystery here. Probably just bad teaching you experienced.
hinkley 1y ago
There was a brief time when I understood how linear algebra was used to render 3d models but that's all gone now. I wonder if these days that would be a better starting point.
- srean 1y ago
  That's just the sad truth of the maximum -- use it or lose it. It can be quite disheartening. The good part is that it becomes easier to pick it up next time around. Happened to me so many times.
imtringued 1y ago
Actually, the calculation of matrix multiplication is extremely easy to understand if you show it done with a row and column vector first. Once your students understand dot products, they will recognize matrix multiplication as the mere pairwise calculation of dot products.
boppo1 1y ago
>Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks. Can you expand on your experience with this? I do some graphics programming so I understand that applying matrix transformations works, and I've seen the 3blue1brown 'matrices are spreadsheets' explanation (luv me sum spreadsheets), but the intuition still isn't really there. The 'incredibly deep "why matrix multiplication looks that way"' is totally lost on me.

Chinjut 1y ago

Why do you say it's practically impossible to motivate matrix multiplication? The motivation is that this represents composition of linear functions, exactly as you follow up by mentioning. It's a disservice to anyone to tell them "Well, that's the way it is" instead of telling them from the start "Look, these represent linear functions. And look, this is how they compose".

andrewla 1y ago
Sure, that's a way to approach it. All you have to do is stay interested in "linear functions" long enough to get there. It's totally possible -- I got there, and so did many many many other people (arguably everyone who has applied mathematics to almost any problem has). But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". I stuck to it because that was what the course taught, and they wouldn't teach me multidimensional calculus without making me learn this stuff first, but it was months and years later when I suddenly found that linear functions were everywhere and I somehow magically had the tools and the knowledge to do stuff with them.
- JadeNB 1y ago
  > But when I was learning linear algebra all I could think was "who cares about linear functions? It's the simplest, dumbest kind of function. In fact, in one dimension it's just multiplication -- that's the only linear function and the class of scalar linear functions is completely specified by the factor that you multiple by". This seems to make it good motivation for an intellectually curious student—"linear functions are the simplest, dumbest kind of function, and yet they still teach us this new and exotic kind of multiplication." That's not how I learned it (I was the kind of obedient student who was interested in a mathematical definition because I was told that I should be), but I can't imagine that I wouldn't have been intrigued by such a presentation!
- ndriscoll 1y ago
  Linear functions are the ones that we can actually wrap our heads around (maybe), and the big trick we have to understand nonlinear problems is to use calculus to be able to understand them in terms of linear ones again. Problems that can't be made linear tend to be exceptionally difficult, so basically any topic you want to learn is going to be calculus+linear algebra because everything else is too hard. The real payoff though is after you do a deep dive and convince yourself there's plenty of theory and all of these interesting examples and then you learn about SVD or spectral theorems and that when you look at things correctly, you see they act independently in each dimension by... just multiplication by a single number. Unclear whether to be overwhelmed or underwhelmed by the revelation. Or perhaps a superposition.
- bananaflag 1y ago
  Yeah, concepts can make a student reject them with passion. I remember in a differential geometry course, when we reached "curves on surfaces", I thought "what stupidity! what are the odds a curve lies exactly on a surface?"
cassepipe 1y ago
Agree.The fact that it's just linear functions is what made it click for me

msgodel 1y ago

I don't know, maybe it's because I read the book on the side in highschool when I was supposed to be doing something else but I really loved Linear Algebra. Once I understood what it was I used matrix operations for everything. Vector spaces are such a powerful abstraction.

srean 1y ago

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these follow the definition of a linear transformation. That rotating a sum is same as summing the rotated(s). [One may also mention that all differentiable functions (from vector to vector) are locally linear.] Then you define adding two linear transformations using vector addition. Next you can define scaling a linear transformation. The point being that the combination can be expressed as linear transformations themself. No need to represent the vectors as R^d, geometric arrows and parallelogram rule would suffice. Finally, one demonstrates composition of linear transformations and the fact that the result itself is a linear transformation. The beautiful reveal is that this addition and composition of linear transformations behave almost the same as addition and multiplication of real numbers. The addition asociates and commutes. The multiplication associates but doesn't necessarily commute. Most strikingly, the operations distributes. It's almost like algebra of real numbers ! Now, when you impose a coordinate system or choose a basis, the students can discover that matrix multiplication rule for themselves over a couple of days of playing with it -- Look, rather than maintaining this long list of linear transformations, I can store it as a single linear transformation in the chosen basis.

cosmic_cheese 1y ago
If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.
- te 1y ago
  codingthematrix.com
- viewtransform 1y ago

dapper_bison17 1y ago

> This "little book" seems to take a fairly standard approach, defining all the boring stuff and leading to Gaussian elimination. The other approach I've seen is to try to lead into it by talking about multi-linear functions and then deriving the notion of bases and matrices at the end. Or trying to start from an application like rotation or Markov chains. Which books or “non-standard” resources would you recommend then, that do a better job?

andrewla 1y ago
I have yet to encounter an approach that is not boring. You just have to power through it. This approach seems as good as any. Once you get to eigenvalues (in my opinion) things start to pick up in terms of seeing that linear spaces are actually interesting. This approach sort of betrays itself when the very first section about scalars has this line: > Vectors are often written vertically in column form, which emphasizes their role in matrix multiplication: This is a big "what?" moment because we don't know why we should care about anything in that sentence. Just call it a convention and later on we can see its utility.
- _mu 1y ago
  Maybe we can petition Paul Lockhart to do a book on Linear Algebra, I would definitely buy it.

Sharlin 1y ago

For anyone who’s interested in graphics programming and/or is a visual learner/thinker, there’s an incredibly motivating and rewarding way to learn the basics of linear algebra. (And affine algebra, which tends to be handwaved away, unfortunately. I’m writing a MSc thesis about this and related topics.)

andrewla 1y ago
To a degree I think this is true, but it requires (at least in my experience) that you have an intrinsic grasp of trigonometry for it to make sense. If you have some complex function analysis and e^itheta then you can skirt the problem for a bit, but if you're like me and have to break out soh-cah-toa whenever you break down a triangle then this method ends up being pretty tedious too.
- Sharlin 1y ago
  I’m not sure what you mean. Beyond rotation matrices, there’s really only trig involved in graphics if you actively want it.
  - andrewla 1y ago
    Maybe I was making unwarranted assumptions about the nature of your way to learn linear algebra. The approaches that I've seen invariably have to produce a sample matrix, and rotation is really the best example. The rotation matrix is going to have sines and cosines, and understanding what that means is not trivial; and even now if you asked me to write a rotation matrix I would have to work it out from scratch. Easy enough to do mechanically but I have no intuitions here even now.
    - Sharlin 1y ago
      Rotation matrices are somewhat mysterious to the uninitiated, but so is matrix multiplication until it "clicks". Whether it ever clicks is a function of the quality of the learning resource (I certainly do not recommend trying to learn linalg via 3D graphics by just dabbling without a good graphics-oriented textbook or tutorial – that usually doesn’t end well). Anyway, I believe that it's perfectly possible to explain rotation matrices so that it "clicks" with a high probability, as long as you understand the basic fact that (cos a, sin a) is the point that you get when you rotate the point (1, 0) by angle a counter-clockwise about the origin (that's basically their definition!) Involving triangles at all is fully optional.

deleted 1y ago

[deleted]

tptacek 1y ago

I didn't think any part of linear algebra was boring. I was hooked from the moment I saw Ax=b => x = b/A. Gaussian elimination is a blast, like an actually-productive Sudoku puzzle, and once you have it down you can blaze through the first 2/3rds of an undergrad linear algebra course. I don't consciously try to gain automaticity with math subjects, but matrix-column multiplication I got pretty quickly and now I just have it. I learned from Strang, for what it's worth, which is basically LU, spaces, QR, then spectral. I am really bad at math, for what it's worth; this is just the one advanced math subject that intuitively clicked for me.

andrewla 1y ago
I haven't looked at Strang's approach. The "x = b / A" is a bit of a gut-punch on first look because my mind immediately tells me all the ways that that does not work. It makes a some sense once I take a second to think about it, and I can see why it would make you want to jump in a little deeper, but matrices being non-commutative makes me cringe at the idea of a division operator which does not very very clearly spell out where it appears in the chain. Ax = b is all well and good, but AxA^-1 = bA^-1 is not meaningful; the application/composition order is very important.
- deleted 1y ago
  [deleted]
dpflan 1y ago
MIT OCW is an amazing resource -- anyone can learn from Strang, which is a goldmine. He also created a course on using Linear Algebra for machine learning: > Linear algebra concepts are key for understanding and creating machine learning algorithms, especially as applied to deep learning and neural networks. This course reviews linear algebra with applications to probability and statistics and optimization–and above all a full explanation of deep learning. - MIT OCW Course: Matrix Methods in Data Analysis, Signal Processing, and Machine Learning (https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/ https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-an...) - The text book website: Linear Algebra and Learning from Data (2019) https://math.mit.edu/~gs/learningfromdata/ https://math.mit.edu/~gs/learningfromdata/ - The Classic Linear Algebra Course: https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...

winwang 1y ago

I hated precalc, but I loved proof-based linear algebra, my first university math course (linalg + multivariate calc). The simplicity(/beauty) of matrix multiplication still irks me though, in the sense of "wow, seriously? when you work it out, it really looks that simple?"

jcranmer 1y ago

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. It's only difficult if you are wedded to a description of matrices and vectors as seas of numbers that you grind your way through without trying to instill a fuller understanding of what those numbers actually mean. The definition makes a lot more sense when you see a matrix as a description of how to convert one sense of basis vectors to another set of basis vectors, and for that, you first need to understand how vectors are described in terms of basis vectors.

nh23423fefe 1y ago
I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.
- jcranmer 1y ago
  If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space. If I write a matrix, say, this: [[1 2] [3 4] [5 6]] What I am doing is describing is a transformation of one vector space into another, by describing how the basis vectors of the first vector space are represented as a linear combination of the basis vectors of the second vector space. Of course, the transformed vectors may not necessarily be a basis of the latter vector space. > The natural motivation of matrices is as representing systems of equations. That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.
  - nh23423fefe 1y ago

joshmarlow 1y ago

The older I get the more convinced I am that "math is not hard; teaching math is hard".

rramadass 1y ago
This is far more truer than most people may realize. Because there is so much to teach/learn, "Modern Mathematics" syllabi has devolved into giving students merely an exposure to all possible mathematical tools in an abstract manner, dis-jointly with no unifying framework, and no motivating examples to explain the need for such mathematics. Most teachers are parrots and have no understanding/insight that they can convey to students and so the system perpetuates itself in a downward spiral. The way to properly teach/learn mathematics is to follow V.I.Arnold's advice i.e. On Teaching Mathematics - https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics https://dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-... Ground all teaching in actual physical phenomena (in the sense of existence with a purpose) and then show the invention/derivation of abstract mathematics to explain such phenomena. Everything is "Applied Mathematics", there is no "Pure Mathematics" which is just another name for "Abstract Mathematics" to generalize methods of application to different and larger classes of problems.
gsinclair 1y ago
As a maths teacher who is interested in (and sufficiently skilled at) programming, I find teaching programming to be very hard, even to interested students. Teaching maths to interested students is not hard (for me).

jameshart 1y ago

What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices. Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.

JadeNB 1y ago
I think that, to be frank, it's a combination of (1) a curriculum developed before it was clear how ubiquitous linear algebra would become, and (2) the fact that it's a lot easier to come up with a standardized assessment for algorithmic calculus than for linear algebra, precisely because linear algebra is both conceptual and proof-based in a way that has been squeezed out of algorithmic calculus. (I use algorithmic calculus to describe the high-school subject, and distinguish it from what in American universities is usually called "analysis," where one finally has the chance to make the acquaintance of the conceptual and proof-based aspects squeezed out of algorithmic calculus.)
bee_rider 1y ago
I think it was a brilliant and evil trick by the linear algebra folks. Start the path at calculus. Naturally, this will lead to differential equations. Trick the engineers into defining everything in terms of differential equations. The engineers will get really annoyed, because solving differential equations is impossible. Then, the mathematicians swoop in with the idea of discretizing everything and using linear algebra to step through it instead. Suddenly they can justify all the million-by-millions matrices they wanted and everybody thinks they are heroes. Engineers will build the giant vector processing machines that they want.
- rramadass 1y ago
  Ha, Ha, True dat :-)
ViscountPenguin 1y ago
That's very strange, where I live linear algebra was a significant portion of the highschool maths curriculum. The actual presentation was terrible, I'll be lucky if I die before having to invert a matrix by hand again, but it was there.

BenFranklin100 1y ago

Probability and statistics falls into that category too. It’s one of the more boring undergraduate math courses, but is mind-boggling useful in the real world. (Basic probability / combinatorics is actually pretty cool, but both tend to be glossed over.)

Sleaker 1y ago

I went through Khan academy on linear algebra a long time ago because I wanted to learn how to write rendering logic, for me I was implementing things as I learned them with immediate feedback for a lot of the material. Was probably the single most useful thing I learned then.

ViscountPenguin 1y ago

I found the university level presentation of "Vector spaces -> Linear functions -> Matrices are isomorphic to linear functions" much more motivating than the rote mechanics I was taught in highschool, but it's hard to see if I would've had that appreciation without being taught the shitty way first.

127 1y ago

I had the opposite experience when learning linear algebra as I was also doing 3D computer graphics at the time. It was super interesting and fun. I guess you just have to find an application for it.

hiAndrewQuinn 1y ago

Linear algebra was such a tedious thing to learn I skipped over it to abstract algebra and doubled back once I had some kind of minimally interesting framework to work with it against. Normally I think this is a foolish way to do things, but sometimes things are just so dull you have to take the hard route to power through at all.

jonahx 1y ago

> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. Short, simple answer to that question by Michael Penn: https://www.youtube.com/watch?v=cc1ivDlZ71U https://www.youtube.com/watch?v=cc1ivDlZ71U Another interesting treatment by Math the World: https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s https://www.youtube.com/watch?v=1_2WXH4ar5Q&t=4s There's no impenetrable mystery here. Probably just bad teaching you experienced.

hinkley 1y ago

There was a brief time when I understood how linear algebra was used to render 3d models but that's all gone now. I wonder if these days that would be a better starting point.

srean 1y ago
That's just the sad truth of the maximum -- use it or lose it. It can be quite disheartening. The good part is that it becomes easier to pick it up next time around. Happened to me so many times.

imtringued 1y ago

Actually, the calculation of matrix multiplication is extremely easy to understand if you show it done with a row and column vector first. Once your students understand dot products, they will recognize matrix multiplication as the mere pairwise calculation of dot products.

boppo1 1y ago

>Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks. Can you expand on your experience with this? I do some graphics programming so I understand that applying matrix transformations works, and I've seen the 3blue1brown 'matrices are spreadsheets' explanation (luv me sum spreadsheets), but the intuition still isn't really there. The 'incredibly deep "why matrix multiplication looks that way"' is totally lost on me.